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Positivity

, Volume 9, Issue 3, pp 415–435 | Cite as

On Normed Lattices and Their Banach Completions

  • A. V. Koldunov
  • A. I. VekslerEmail author
Article
  • 57 Downloads

Abstract

It is known that the Banach completion Y = bX of a normed lattice X need not preserve the properties to be Dedekind complete or σ-Dedekind complete. In this paper it is proved that the countable interpolation property and the property to be sequentially order complete are preserved under the Banach completion. To prove this results we found some sufficient conditions (which are close to necessary ones) on X which secure for Y to have the countable interpolation property and (respectively) to be sequentially order complete. These conditions are obtained with the help of the newly developed techniques based on representations of normed lattices. It is well known that order continuity, and σ-order continuity of a norm are preserved under the Banach completion. Here necessary and sufficient conditions on X to secure these properties in Y are discussed.

Keywords

normed lattice Banach completion countable interpolation property sequential order completeness principle compact spaces norm functions X-principal ideal XPI-property order continuity σ-order continuity 

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Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.St-Petersburg Federal Pedagogic UniversitySt-PetersburgRussia
  2. 2.St-Petesburg Federal University of Technology and DesignSt-PetersburgRussia

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